Differential entropy

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The formal analogue of the concept of entropy for random variables having distribution densities. The differential entropy of a random variable defined on some probability space , assuming values in an -dimensional Euclidean space and having distribution density , , is given by the formula

where is assumed to be equal to zero. Thus, the differential entropy coincides with the entropy of the measure with respect to the Lebesgue measure , where is the distribution of .

The concept of the differential entropy proves useful in computing various information-theoretic characteristics, in the first place the mutual amount of information (cf. Information, amount of) of two random vectors and . If , and (i.e. the differential entropy of the pair ) are finite, the following formula is valid:

The following two properties of the differential entropy are worthy of mention: 1) as distinct from the ordinary entropy, the differential entropy is not covariant with respect to a change in the coordinate system and may assume negative values; and 2) let be the discretization of an -dimensional random variable having a density, with steps of ; then for the entropy the formula

is valid as . Thus, as . The principal term of the asymptotics of depends on the dimension of the space of values of . The differential entropy defines the term next in order of the asymptotic expansion independent of and it is the first term involving a dependence on the actual nature of the distribution of .


[1] I.M. Gel'fand, A.N. Kolmogorov, A.M. Yaglom, "The amount of information in, and entropy of, continuous distributions" , Proc. 3-rd All-Union Math. Congress , 3 , Moscow (1958) pp. 300–320 (In Russian)
[2] A. Rényi, "Wahrscheinlichkeitsrechnung" , Deutsch. Verlag Wissenschaft. (1962)
How to Cite This Entry:
Differential entropy. R.L. DobrushinV.V. Prelov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098